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1/48 Sajid Ullah BUTT Conception et modélisation d'un montage de fabrication pour le balançage optimisé d'une famille de pièces Arts et Métiers ParisTech.

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Presentation on theme: "1/48 Sajid Ullah BUTT Conception et modélisation d'un montage de fabrication pour le balançage optimisé d'une famille de pièces Arts et Métiers ParisTech."— Presentation transcript:

1 1/48 Sajid Ullah BUTT Conception et modélisation d'un montage de fabrication pour le balançage optimisé d'une famille de pièces Arts et Métiers ParisTech - Centre de Metz Laboratoire de Conception Fabrication Commande EA 4495 Jury M. Cornel Mihai NICOLESCU, Professeur, KTH, Stockholm, SwedenRapporteur M. Jean-François RIGAL, Professeur, LAMCOS, INSA Lyon, France Rapporteur M. Henri PARIS, Professeur, G.SCOP, Université Joseph Fourier, Grenoble, FranceExaminateur M. Jean-François ANTOINE, Maitre de conférences, IUT de Nancy Brabois, FranceCo-directeur de thèse M. Patrick MARTIN, Professeur, LCFC, Arts et Métiers ParisTech, Metz, FranceDirecteur de thèse

2 2/48 Presentation layout Sajid Ullah BUTT, PhD Defense 5 July 2012 Case Study Context Kinematic Model Mechanical Model Conclusion and Perspectives

3 3/48 Presentation layout Sajid Ullah BUTT, PhD Defense 5 July 2012 Case Study Context Kinematic Model Mechanical Model Conclusion and Perspectives C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION

4 4/48 R Initial surface R Context Optimal balancing The final product should have a minimum allowance for better machining In case of perfect positioning, minimum rough part radius should have to be r + h is the positioning error between the final product and the rough parts central axis The minimum radius of the rough part has to be R for a good machining operation More positioning error will increase the material waste C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION h Allowance > Min chip thickness h Sajid Ullah BUTT, PhD Defense 5 July 2012 r r Final part L

5 5/48 Part Pallet Column Base Spindle Kinematics defects Locators placement Geometric/form defects Deformation due to forces Context Workpiece/machine tool Positioning error Variation among the parts of the same part family cause the positioning error during fixturing Positioning error of the workpiece affects the quality of the final product Tool wear Effect of heat NC Code errors C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Tool Sajid Ullah BUTT, PhD Defense 5 July 2012

6 6/48 Possible placement of locators Placement of locators Block the 6-DOFs of the part Placement procedure Choose the locating surfaces taking into account the constraints of accessibility, load, external force and movements (Somashekar 2002) Select the locators configurations (3-2-1, 3-2-1C, etc.) Choose the locators positions for the part stability (Roy & Liao, 2002; Zirmi et al. 2009) Positioning errors C (H. Paris, 1995) C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Y -Y Z -XX -Z 0 Sajid Ullah BUTT, PhD Defense 5 July 2012

7 7/48 Geometrical and form defects When workpiece is placed directly on the locators Local geometrical defects cause the orientation error The orientation error have more effect on the final product quality than the translation error (Asante, 2009) Positioning errors C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Sajid Ullah BUTT, PhD Defense 5 July 2012

8 8/48 F Deformation of locators under external load The locators and their contacts deform under clamping and machining forces Deformation depends upon the stiffness of the locators Hertz contact theory may be applied to calculate the contact deformation Locators deformations induce the workpiece displacement Zero contact deformation Including contact deformation Positioning errors C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Z Y X F F F Sajid Ullah BUTT, PhD Defense 5 July 2012

9 9/48 Positioning errors Machine tool/kinematic chain defects Machine tool position uncertainty Kinematic chain Kinematic defects increase with the increase the number of machine axes Other Defects Defects due to heat generation NC code defects Tool wear C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Sajid Ullah BUTT, PhD Defense 5 July 2012 How to Compensate these errors? How to Compensate these errors?

10 10/48 Error compensation Existing methods Changing the part program Easiest way (Ramesh et al. 2000) Orientation of the machine tool Disadvantages Need 4 or 5 axis machines Very expensive for the existing production line Actual position Ideal position Compensated position (Zhu et al. 2012) C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Sajid Ullah BUTT, PhD Defense 5 July 2012 Base Part Tool Column Pallet Part program Tool orientation

11 11/48 Base Tool Column Part Baseplate Part Baseplate Pallet 6 DOF repositioning Error compensation C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Our proposal A 6-DOF workpiece repositioning system is proposed A baseplate is introduced to avoid the positioning error caused by the geometrical defects Repositioning is performed through the positioning of the 6 locators

12 12/48 Develop a fixturing system which can –Hold custom single and complex parts –Perform 6-DOF Repositioning of the part at desired position –Added to Single machine unit or production/assembly line –Minimum modifications on existing production line Part Baseplate Tool Part Baseplate Conveyor Previous Workstation Reconfigurable Pallet Base Column Pallet Objective C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Single machine unit Production/Assembly line Sajid Ullah BUTT, PhD Defense 5 July 2012 MeasureCalculateCompensate Part Baseplate Reconfigurable Pallet Part Baseplate

13 13/48 Measure Workpiece geometrical errors Offline: using CMM Online: Integrated sensors Calculate The advancement of locators required to compensate the errors using Homogeneous Transformation Matrices and Large displacements (Kinematic model) The errors due to deformation of elastic elements under load using Small Displacement hypothesis (Mechanical model) Compensate Through the axial advancement of 6 locators Geometrical errors Mechanical errors Objective C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Sajid Ullah BUTT, PhD Defense 5 July 2012 Error compensation principle

14 14/48 Presentation layout Sajid Ullah BUTT, PhD Defense 5 July 2012 Context Positioning errors Compensation Objectives Workpiece repositioning through locators All the elements rigid Proposed fixturing system Analytical formulation Large displacements and Homogeneous Transformation Matrices Deformation of elastic elements and rigid body displacement of part on fixture under load Deformation of locators and contacts Lagrangian formulation Negligible friction, Small Displacements Convergence of non- linear contact deformation Work realized Conclusion Future work Case Study Context Kinematic Model Mechanical Model Conclusion and Perspectives C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION

15 15/48 Objective Repositioning of a part of a part family placed roughly with a precision of some millimeters Components –Part (Hip prosthesis) –Baseplate (Cuboid) –6-Locators (Axial movement) –Pallet All the elements are rigid X Z Y O ( Machine/Pallet reference) P Baseplate part Kinematic model C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Sajid Ullah BUTT, PhD Defense 5 July 2012 Part Baseplate

16 16/48 [P PF ] = [P bb ] XFXF XFXF XbXb XbXb XOXO XOXO XbXb XbXb XPXP XPXP [P PF ] [P bP ] [P bF ]=[P bP ] [P Ob ] Correction [P PF ] Error to be corrected Rigid link Baseplate correction through locators Initial baseplate placement on the locators Formulation X Z Y O ( Machine/Pallet reference) Y3Y3 Z3Z3 X3X3 b P XPXP ZPZP YPYP Baseplate part C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Sajid Ullah BUTT, PhD Defense 5 July 2012 Baseplate Surface normals

17 17/48 CPT 12/14 Hip Prosthesis Zimmer Initial Position of the workpiece (2D Simulation) Min Material (Chebyshev) RMS C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION P Sajid Ullah BUTT, PhD Defense 5 July 2012 Y X Point P (intersection of two centerlines) Definition of a plane Simulation in 2D Stem Neck

18 18/48 P Min Material (Chebyshev) RMS Point P (intersection of two centerlines) Definition of a plane Simulation in 2D C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Sajid Ullah BUTT, PhD Defense 5 July 2012 Y X Initial Position of the workpiece (2D Simulation) Stem Neck

19 19/48 Z X 1* 2* Final calculated Position Calculating point of contact in axis Position calculation in 3D Compensation of errors C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Sajid Ullah BUTT, PhD Defense 5 July D Schematic explanation

20 20/48 Simulation procedure: CAD Modeling C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Inverse impression of the workpiece for the simulation of cutting tool path D=3mm Cavity with original workpiece dimensions Baseplate Pallet Workpiece Sajid Ullah BUTT, PhD Defense 5 July 2012 Boolean Operation

21 21/48 Case study Initial Data Final required part position Calculated positions of locators C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Gray: Machined surface Orange: Rough surface Sajid Ullah BUTT, PhD Defense 5 July 2012

22 22/48 Simulated machining Positioning error of workpiece after correction Case study Calculated positions of locators Final Product C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Sajid Ullah BUTT, PhD Defense 5 July 2012 Calculated positions of locators Positioning error of workpiece after second side correction

23 23/48 Robustness of model/Sensitivity Analysis Use of Plucker matrix Precision of workpiece displacement as a function of locators positioning precision Position uncertainty = Geometrical uncertainty + uncertainty due to temperature C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Sensitivity analysis Sajid Ullah BUTT, PhD Defense 5 July 2012 Worst case Precision of locator

24 24/48 Presentation layout Sajid Ullah BUTT, PhD Defense 5 July 2012 Context Positioning errors Compensation Objectives Workpiece repositioning through locators All the elements rigid Proposed fixturing system Analytical formulation Large displacements and Homogeneous Transformation Matrices Deformation of elastic elements and rigid body displacement of part on fixture under load Deformation of locators and contacts Lagrangian formulation Negligible friction, Small Displacements Convergence of non- linear contact deformation Work realized Conclusion Future work Case Study Context Kinematic Model Mechanical Model Conclusion and Perspectives C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION

25 25/48 Mechanical model Clamping and machining forces and moments Elements designed to be rigid –Workpiece baseplate assembly –Mass elements Elastic elements –Locators (body and contact) –Baseplate at contacts –Clamps with imposed external displacements Small displacement hypothesis Friction neglected Effects of heat neglected No slippage of clamps at contact {X E } 2 Z X Y P [K] 1 [K] 2 [K] 3 [K] 4 [K] 5 [K] 6 {X E } 1 [K E ] 2 [K E ] 1 f | T, F | C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Sajid Ullah BUTT, PhD Defense 5 July 2012

26 26/48 Formulation X F* X b* XFXF XFXF XbXb XbXb XOXO XOXO P bF =P bP P ob* P b*F* =P bP P Ob P b*b P FF* Rigid link Correction through locators Error of the workpiece under load Machining forces and their displacement P bb* =P FF* Clamping forces and their displacements Initial baseplate locating under load Correction {X E } 2 Z X Y P [K] 1 [K] 2 [K] 3 [K] 4 [K] 5 [K] 6 {X E } 1 [K E ] 2 [K E ] 1 f | T, F | C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Sajid Ullah BUTT, PhD Defense 5 July 2012

27 27/48 Formulation Work done by external force Lagrangian Equation Z X Y K3K3 6 K6K6 P ZPZP XPXP YPYP 5 K5K5 4 K4K4 K2K2 K1K1 Baseplate K E1 X E1 K E2 X E2 Part F T Machine/Pallet Reference {ΔX,ΔY, ΔY} T : Linear displacement vector of point P {Δα,Δβ, Δγ} T : Angular displacement vector of point P {F}: Force vector {T}: Moment vector C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION U: Potential energy of the system T: Kinetic energy of the system W:Work done by the external forces q i : Generalized coordinates Sajid Ullah BUTT, PhD Defense 5 July 2012

28 28/48 Formulation Potential Energy: U Lagrangian Equation Kinetic Energy: T Z X Y K3K3 6 K6K6 P ZPZP XPXP YPYP 5 K5K5 4 K4K4 K2K2 K1K1 Baseplate K E1 X E1 K E2 X E2 Part F T Machine/Pallet Reference Locators Clamps C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION (Lalanne et al. 1986) Sajid Ullah BUTT, PhD Defense 5 July 2012

29 29/48 Locator Pin 68mm 20mm Locator model Screw-nut Wedge-slop locator Rotation of knob causes axial movement of locator Slope 1:2 Screw M6x1 Locator diameter: 20mm Length of locator: 68mm Sphere radius: 20mm C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Perforated plate Pitch = 40mm Sajid Ullah BUTT, PhD Defense 5 July 2012

30 30/ Initial position of the surface Final position of the surface Locator axis Z Y X Deformed locator at the position having minimum potential energy Formulation (Zero Friction) C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION 3 4 Bending + Shear Compression Minimum energy (Menabreas theorem)Potential energy of locators Locators Stiffness Matrix Sajid Ullah BUTT, PhD Defense 5 July 2012

31 31/48 Final position of contact surface after only locator deformation Final position of contact surface after locator and contact deformation Z Y X Zero contact deformation Including contact deformation Formulation Deformation of contact (Hertz contact theory) C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Sajid Ullah BUTT, PhD Defense 5 July 2012

32 32/48 Formulation (Iterations procedure) New stiffness matrices of each locator [K] i Stiffness matrix [K], overall displacement vector {X} and natural frequencies of the system using Lagrangian Deformation of each locator{δ} i Potential energy calculations Deformation and stiffness of i th locator body (δ L, K L ) i Deformation and stiffness of i th contact (δ C, K C ) i Deformation and stiffness of i th contact (δ C, K C ) i Overall stiffness of each locator and displacement vector of the workpiece ({X new } ) using inverse Plucker [K New ] i = {F} i /{δ New } i T {X New }=[Plu] -1 {δ Plu } Overall stiffness of each locator and displacement vector of the workpiece ({X new } ) using inverse Plucker [K New ] i = {F} i /{δ New } i T {X New }=[Plu] -1 {δ Plu } Final deformation/displacement vector and stiffness matrix of each locator and the fixturing system No Yes/STOP Kinetic energy and Work done Kinetic energy and Work done [K] i =[K New ] i {X}= {X New } +gain*({X} - {X new } ) {δ New } i = {δ C } i +{δ L } i {X New } [K] i C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Z Y X 3D stiffness matrix of the locator body [K L ] i [K C ] i {F} i 3D stiffness matrix of the contact 3D equivalent stiffness matrix {F} i [K New ] i Force vector on the i th locator {F} i Sajid Ullah BUTT, PhD Defense 5 July 2012 Gain

33 33/48 Y X Z P O {X E } 2 Z X Y P [K] 1 [K] 2 [K] 3 [K] 4 [K] 5 [K] 6 {X E } 1 [K E ] 2 [K E ] 1 f Case study | T, F | C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Sajid Ullah BUTT, PhD Defense 5 July 2012

34 34/48 Clamps Baseplate (made of steel) {X E } 2 Z X Y P [K] 1 [K] 2 [K] 3 [K] 4 [K] 5 [K] 6 {X E } 1 [K E ] 2 [K E ] 1 f locators-baseplate contacting points Contacting points of clamps Case study (Input) Locator stiffness | T, F | 68mm 20mm C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Sajid Ullah BUTT, PhD Defense 5 July 2012 Extracted from CAD model

35 35/48 Processed material Prosthesis material : M30NW (X4CrNiMoN21) ISO equivalent material: M3.2.C.AQ (Stainless steel Cast, Annealed quenched) –Stainless steel 316LN (X2CrNiMoN18-13, Sandvik technical guide 2011) –σ = 880MPa Cutting condition –Tool : CoroMill 216 ball nose endmill of 3 mm in diameter (2 teeth) –DOC, a p : 0.5 mm, Feed per tooth, Fz : 0.03 mm –Cutting speed, Vc: 75 m/min, Spindle speed, N : 8000 RPM Machining forces –Tangential force (Sandvik technical guide 2011) –Repulsive forces (Pruvot, 1993) Case study (Input) C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Sajid Ullah BUTT, PhD Defense 5 July 2012

36 36/48 Results without considering contact deformation Case study (Results) Results with considering contact deformation Natural frequencies of the systemError compensation C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Sajid Ullah BUTT, PhD Defense 5 July 2012 Tool Excitation frequency=837 rad/sec

37 37/48 Convergence of displacement vector μRad, μm No of iterations % Error C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Sajid Ullah BUTT, PhD Defense 5 July 2012

38 38/48 Effect of gain on convergence μRad No of iterations C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Convergence of parameter slowest parameter γ Sajid Ullah BUTT, PhD Defense 5 July 2012

39 39/48 Rough contacts C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Baseplate-locator equivalent RMS roughness = 0.8 E -6 m k C /k H RMS Roughness (m) 1N 10N 100N 1kN Comparison between ideal and rough surface contacts (Bahrami et al. 2005) Sajid Ullah BUTT, PhD Defense 5 July % decrease

40 40/ locator configuration {X E } 2 Z X Y P [K] 1 [K] 2 [K] 3 [K] 4 [K] 5 [K] 6 {X E } 1 [K E ] 2 [K E ] 1 f | T, F | {X E } 2 Z X Y P [K] 1 [K] 2 [K] 3 [K] 4 [K] 5 [K] 6 {X E } 1 [K E ] 2 [K E ] 1 f [K] 7 [K] 8 | T, F | System stiffness Comparison between and C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Sajid Ullah BUTT, PhD Defense 5 July 2012

41 41/48 Presentation layout Sajid Ullah BUTT, PhD Defense 5 July 2012 Context Positioning errors Compensation Objectives Workpiece repositioning through locators All the elements rigid Proposed fixturing system Analytical formulation Large displacements and Homogeneous Transformation Matrices Deformation of elastic elements and rigid body displacement of part on fixture under load Deformation of locators and contacts Lagrangian formulation Negligible friction, Small Displacements Convergence of non- linear contact deformation Work realized Conclusion Future work Case Study Context Kinematic Model Mechanical Model Conclusion and Perspectives C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION

42 42/48 Conclusion High quality baseplate is introduced Compensation is performed through advancement of locators (kinematic model) Deformation of each locator is calculated and its contact with baseplate under load (mechanical model) Also its resultant rigid body displacement of the workpiece is calculated (positioning error) Compensation is performed using kinematic model C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Base Tool Column Part Baseplate Part Baseplate Pallet Kinematics defects Locators placement Geometric/form defectsDeformation due to forces Sajid Ullah BUTT, PhD Defense 5 July DOF part repositioning

43 43/48 Conclusion 6 DOF positioning using kinematic model –3-2-1 locating configuration is used –All elements are considered rigid –Error compensation is performed through the axial translation of 6-locators using HTM and LD –Validated on a case study of repositioning the hip prosthesis –Sensitivity analysis are carried out C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Sajid Ullah BUTT, PhD Defense 5 July 2012

44 44/48 Conclusion Mechanical modeling of the fixturing system –The analytical model is developed –Deformation of locators under load is calculated –Workpiece-baseplate assembly is designed to be rigid –Locators, clamps and locator-baseplate contacts are assumed deformable –Small displacement hypothesis are used –Lagrangian formulation is used to calculate the mass & stiffness matrices and workpiece displacement vector –Non-linear behavior of locator-baseplate contact is linearized –Demonstrated on a case study of locating configuration and compared with configuration of locators C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Sajid Ullah BUTT, PhD Defense 5 July 2012

45 45/48 Analytical modeling gives very quick result as compared to numerical modeling The proposed mechanical model can easily be applied to more complex problems with multiple loads, different orientations and stiffness of locators and clamps The proposed fixturing system allows precise positioning of the workpiece at each workstation without the need of 4 or 5 axis machines or modifying the existing workstations Reduce dimensional errors, machining allowances and thus the material removal by uniformly centering the rough part to the required part Consequently, it reduces the material waste Large parts could also be repositioned during assembling Conclusion C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Sajid Ullah BUTT, PhD Defense 5 July 2012

46 46/48 The positioning error due to heat/temperature change can be introduced Construction of the fixturing system for validating the analytical model. We could not construct the model because of time and cost associated with precise part production The mechanical model calculates the deformation of locators as the result of an instantaneous force at a point. The model should be developed to simulate the whole tool path Limitations and Future work C ONTEXT K INEMATIC M ODEL M ECHANICAL MODEL C ONCLUSION Sajid Ullah BUTT, PhD Defense 5 July 2012

47 47/48 Scientific activities Poster Presentation Conception, Modélisation et Réalisation d'un montage modulaire rapide destine a la Fabrication Mécanique J2A Arts et Métiers ParisTech, 8 th -9 th June 2010 (Poster Presentation) National Colloquium S. U. Butt, J. F. Antoine, P. Martin, Mechanical model for control of 6 DOF repositioning system, 12th National Colloquium, AIP Primeca, Mont-Dore 29th March to 1st April 2011 Scientific Publications S. U. Butt, J. F. Antoine, P. Martin, An analytical model for the repositioning of 6 DOF repositioning system, Journal of Mechanics and Industry (Accepted June 2012 ) S. U. Butt, J. F. Antoine, P. Martin, An analytical stiffness model for spherical rough contacts, Asian International Journal of Science and Technology in Production and Manufacturing Engineering (Submitted June 2012) Sajid Ullah BUTT, PhD Defense 5 July 2012

48 48 f Thanks for your attention

49 49/48 Measuring locator positioning accuracy Measuring the accuracy of the locator movement Designed slope 1:2 Obtained av. slope:

50 50/48 Measuring locator positioning accuracy

51 51/48 Backlash in the advancement

52 52/48 Position of the workpiece Z 0 =Z 1 P YPR Transformation (Yaw, Pitch,Roll) Z Y X O βγ Y0Y0 XPXP Z 0 =Z 1 γ β X0X0 Y 2 =Y P X 1 =X 2 Z2Z2 ZPZP Y1Y1 P(x P, y P, z P ) α α

53 53 Baseplate position in machine X Y Z u w v b


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